**4. Scaling Group Analysis**

The governing boundary layer flow and heat transfer problem in the form of partial differential equations (PDEs) is complex and hard to solve by means of mathematical software. Therefore, it needs to be reduced to a simpler form so that it can be solved. Suitable similarity variables can facilitate the transformation and, at this point, scaling group analysis is required to form the specified similarity transformations for the present problem. The newly formed similarity variable will then transform the PDEs to a system of ordinary differential equations (ODEs), and the model can be solved by the desired mathematical software. Therefore, the following scaling group of transformations *G* is introduced:

$$\begin{aligned} \text{G}: \text{x}^\* &= \text{x} \text{G}^{\omega\_1}, & y^\* &= y \text{G}^{\omega\_2}, & \psi^\* &= \psi \text{G}^{\omega\_3}, & \sigma^\* &= \sigma \text{G}^{\omega\_4},\\ \text{G}^\* &= \theta \text{G}^{\omega\_5}, & u\_\varepsilon^\* &= u\_\varepsilon \text{G}^{\omega\_6}, & u\_1^\* &= u\_1 \text{G}^{\omega\_7}, & m^\* &= m \text{G}^{\omega\_8}, \end{aligned} \tag{14}$$

where ω*i* are constants to be determined in which *i* = 1, ... 8. The transformation *G* is the transformation point which transforms the (*x*, *y*,ψ, σ, θ, *ue*, *u*1, *<sup>m</sup>*,) coordinates to the new coordinates *<sup>x</sup>*<sup>∗</sup>, *y*<sup>∗</sup>,ψ<sup>∗</sup>, <sup>σ</sup><sup>∗</sup>, θ<sup>∗</sup>, *<sup>u</sup>*<sup>∗</sup>*e*, *<sup>u</sup>*<sup>∗</sup>1, *<sup>m</sup>*<sup>∗</sup>.

Next, the substitution of (14) into Equations (9)–(11) yields the following expressions:

$$\begin{split} \frac{A\_{1}}{A\_{2}} \mathbf{G}^{[2\omega\_{3} - 2\omega\_{2} - \omega\_{1}]} \Big( \frac{\partial \psi^{\*}}{\partial y^{\*}} \frac{\partial^{2} \psi^{\*}}{\partial x^{\*} \partial y^{\*}} - \frac{\partial \psi^{\*}}{\partial x^{\*}} \frac{\partial^{2} \psi^{\*}}{\partial y^{\*2}} \Big) &= G^{[\omega\_{3} - 3\omega\_{2}]} \Big( \frac{\partial^{3} \psi^{\*}}{\partial y^{\*3}} \Big) \\ &+ \frac{A\_{1}}{A\_{2}} \frac{\partial \omega}{\partial x} G^{[\omega\_{8} + 3\omega\_{3} - 7\omega\_{2}]} \Big[ m^{\*} \left( \frac{\partial^{2} \psi^{\*}}{\partial y^{\*2}} \right)^{2} \frac{\partial^{3} \psi^{\*}}{\partial y^{\*3}} \Big] + G^{[2\omega\_{6} - \omega\_{1}]} \left. u^{\*}\_{e} \frac{du^{\*}\_{\mathbf{t}}}{d\mathbf{t}^{\star}} \right. \\ &\left. - \frac{B\_{0}^{2}}{\rho\_{bf} d} \frac{1}{A\_{2}} \Big[ G^{[\omega\_{3} + \omega\_{4} - \omega\_{2}]} \Big( \sigma^{\*} \frac{\partial \psi^{\*}}{\partial y^{\*}} \right) - G^{[\omega\_{4} + \omega\_{6}]} \big( \sigma^{\*} u^{\*}\_{e} \Big) \Big]. \end{split} \tag{15}$$

$$G^{\left[\omega\_{3}+\omega\_{5}-\omega\_{1}-\omega\_{2}\right]}\left(\frac{\partial\psi^{\*}}{\partial y^{\*}}\frac{\partial\theta^{\*}}{\partial\mathbf{x}^{\*}}-\frac{\partial\psi^{\*}}{\partial\mathbf{x}^{\*}}\frac{\partial\theta^{\*}}{\partial y^{\*}}\right) = \frac{1}{Pr}G^{\left[\omega\_{5}-2\omega\_{2}\right]}\left(\frac{\partial^{2}\theta^{\*}}{\partial y^{\*2}}\right) + \frac{4}{3}\frac{Rd}{Pr}G^{\left[\omega\_{5}-2\omega\_{2}\right]}\left(\frac{\partial^{2}\theta^{\*}}{\partial y^{\*2}}\right),\tag{16}$$

along with the boundary conditions:

$$\begin{cases} G^{[\omega\_3-\omega\_2]} \Big( \frac{\partial \psi^\*}{\partial y^\*} \Big) = G^{[\omega\_T + \frac{2}{3}\omega\_1]} \Big( \frac{u\_1^\*}{u\_0} \mathbf{x}^\* \frac{2}{3} \Big),\\ G^{[\omega\_3-\omega\_1]} \Big( \frac{\partial \psi^\*}{\partial x} \Big) = -\frac{v\_1}{\sqrt{a^{\omega\_b}}} G^{[-\frac{2}{3}\omega\_1]} \mathbf{x}^{\*-\frac{2}{3}},\ G^{[a\omega\_3]} \theta^\* = G^{[\frac{2}{3}\omega\_1]} \Big( \mathbf{x}^\* \frac{2}{5} \Big) \quad \text{at} \quad y = 0\\ G^{[\omega\_3-\omega\_2]} \Big( \frac{\partial \psi^\*}{\partial y^\*} \Big) \to G^{[\omega\_b]} (u\_c^\*) \quad \text{as} \quad y \to \infty. \end{cases} \tag{17}$$

To retain the invariance of the system under *G*, the parameters defined in Equation (14), the following relations must hold:

$$\begin{aligned} 2\omega\_3 - 2\omega\_2 - \omega\_1 &= \omega\_3 - 3\omega\_2 = 3\omega\_3 - 7\omega\_2 + \omega\_8 = 2\omega\_6 - \omega\_1 = \omega\_3 - \omega\_2 + \omega\_4 = \omega\_4 + \omega\_6 \\ 2 &= \omega\omega + \omega\mathfrak{z} - \omega\mathfrak{y} - \omega\mathfrak{z} = \omega\mathfrak{z} - 2\omega\mathfrak{z}. \end{aligned} \tag{18}$$

From the boundary conditions of Equations (17), we also obtain the following relations among the parameters:

$$
\omega\_3 - \omega\_2 = \omega\_7 + \frac{2}{5}\omega\_1, \quad \omega\_3 - \omega\_1 = -\frac{2}{5}\omega\_1, \quad \omega\_5 = \frac{2}{5}\omega\_1, \quad \omega\_3 - \omega\_2 = \omega\_6. \tag{19}
$$

The absolute invariant can be determined by eliminating the parameter *G* of the group and hence Equations (18) and (19) provide the following expressions:

$$\begin{array}{llll}\omega\_2 = \frac{2}{5}\omega\_1, & \omega\_3 = \frac{2}{5}\omega\_1, & \omega\_4 = -\frac{4}{5}\omega\_1, & \omega\_5 = \frac{2}{5}\omega\_1, \\ \omega\_6 = \frac{1}{5}\omega\_1, & \omega\_7 = -\frac{1}{5}\omega\_1, & \omega\_8 = \frac{2}{5}\omega\_1. \end{array} \tag{20}$$

From Equations (13), (14), and (20), we achieve the absolute invariants under the group *G* similarity transformations as follows:

$$\begin{cases} \eta = \frac{y}{\frac{\sigma}{2}}, \quad \psi = \mathbf{x}^{\frac{3}{5}} f(\eta), \quad \sigma = \sigma\_0 \mathbf{x}^{-\frac{4}{5}}, \quad \theta = \theta\_0(\eta) \text{ x}^{\frac{2}{5}},\\ u\_\varepsilon = (u\_\varepsilon)\_0 \mathbf{x}^{\frac{1}{5}}, \quad u\_1 = (u\_1)\_0 \mathbf{x}^{-\frac{1}{5}}, \quad m = m\_0 \text{ x}^{\frac{2}{5}}. \end{cases} \tag{21}$$

The similarity transformations in Equation (21) are new and, by employing them in the governing boundary layer equations of Equations (9) and (11), the reduced version of the model in the form of ordinary di fferential equations can be attained as follows while satisfying Equation (9):

$$\frac{A\_1}{A\_2} f'' \left[ 1 + \frac{m\_0 D \epsilon}{2} \left( f'' \right)^2 \right] - \frac{1}{5} (f')^2 + \frac{3}{5} f \, f'' + \frac{1}{5} - \frac{M}{A\_2} (1 - f') = 0,\tag{22}$$

$$\left(A\_4 + \frac{4}{3}Rd\right)\theta'' - \frac{2}{5}A\_3 \text{Pr} \, f' \, \theta + \frac{3}{5}A\_3 \text{Pr} \, f \, \theta' = 0,\tag{23}$$

with the associated boundary conditions:

$$f(0) = \frac{5}{3} f\_{\overline{w}}, \quad f'(0) = \varepsilon, \quad \theta(0) = 1, \quad f'(\infty) = 1, \quad \theta(\infty) = 0. \tag{24}$$

Here ε = (*<sup>u</sup>*1)0/*<sup>u</sup>*<sup>0</sup> is the stretching/shrinking parameter, where ε > 0 indicates the stretching sheet, ε = 0 specifies the stationary sheet, and ε < 0 represents the state of shrinking sheet. Furthermore, *fw* = −*<sup>v</sup>*1/ √<sup>ν</sup>*b f a* is the constant mass transfer parameter, and *fw* > 0 typifies the suction e ffect at the surface of the moving sheet and *fw* < 0 epitomizes the injection state. For simplicity, we choose (*ue*)0 = 1. The power-law index is denoted by *m*0; when *m*0 = 0, the reduced model imitates the Newtonian fluid behavior. Moreover, when *m*0 = 1, the model is reduced to the Eyring model, whereas the present model in Equations (22)–(24) also predicts specifically the pseudo-plastic (shear thinning) and dilatant (shear thickening) fluid properties when *m*0 < 0 and *m*0 > 0, respectively.

*Mathematics* **2020**, *8*, 1430

The physical quantities of interest in the present work are the local skin friction coefficient -*Cf x* and the local Nusselt number (*Nux*) which are defined as follows:

$$C\_{f\overline{\chi}} = \frac{\pi\_{\text{uv}}}{\rho\_{bf}\overline{u}\_{\text{c}}^{2}}, \quad Nu\_{\overline{\chi}} = \frac{\overline{\chi}q\_{\text{uv}}}{k\_{bf}(T\_{\text{uv}} - T\_{\text{oc}})} ,\tag{25}$$

where τ*w* is the wall shear stress and *qw* is the heat flux at the surface of the sheet, and can be further defined as [34]:

$$\tau\_w = \mu\_{nf} \left\{ 1 + \frac{mE^2}{6} \left[ 2 \frac{\partial \overline{u}}{\partial \overline{x}} \right]\_{\overline{y}=0}^2 + \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)\_{\overline{y}=0} \right\} \left| \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)\_{\overline{y}=0} \right. \\ \qquad q\_w = -k\_{nf} \left( \frac{\partial T}{\partial \overline{y}} \right)\_{\overline{y}=0} . \tag{26}$$

The reduced skin friction coefficient Re1/2 *x Cf xx*1/5 and the local Nusselt number Re−1/2 *x Nux x*<sup>−</sup>1/5 can be obtained using the similarity transformations of Equation (21) and the expressions in Equations (25) and (26) as follows:

$$\begin{array}{l} \mathrm{C}\_{f\text{x}}\mathrm{Re}\_{\frac{1}{\mathfrak{X}}}^{1/2} = \sqrt{A\_1A\_2}f^{\prime\prime}(0) + \frac{m\_0}{6\sqrt{A\_1A\_2}}\mathrm{De}[f^{\prime\prime}(0)]^{\frac{\mathfrak{Z}}{\mathfrak{Z}}},\\ \mathrm{Re}\_{\overline{\mathfrak{X}}}^{-1/2}\mathrm{Nu}\_{\overline{\mathfrak{X}}}\mathrm{x}^{-2/5} = -\frac{1}{\sqrt{A\_1A\_2}}\left(1 + \frac{1}{A\_4}\frac{4}{3}\mathrm{R}d\right)\frac{\vartheta^{\prime}(0)}{\vartheta(0)},\end{array} \tag{27}$$

where Re*x* = *uex*/<sup>ν</sup>*b f* denotes the local Reynolds number.
